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Theorem opprval 18393
Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprval.1 𝐵 = (Base‘𝑅)
opprval.2 · = (.r𝑅)
opprval.3 𝑂 = (oppr𝑅)
Assertion
Ref Expression
opprval 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)

Proof of Theorem opprval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 opprval.3 . 2 𝑂 = (oppr𝑅)
2 id 22 . . . . 5 (𝑥 = 𝑅𝑥 = 𝑅)
3 fveq2 6088 . . . . . . . 8 (𝑥 = 𝑅 → (.r𝑥) = (.r𝑅))
4 opprval.2 . . . . . . . 8 · = (.r𝑅)
53, 4syl6eqr 2661 . . . . . . 7 (𝑥 = 𝑅 → (.r𝑥) = · )
65tposeqd 7219 . . . . . 6 (𝑥 = 𝑅 → tpos (.r𝑥) = tpos · )
76opeq2d 4341 . . . . 5 (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
82, 7oveq12d 6545 . . . 4 (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
9 df-oppr 18392 . . . 4 oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩))
10 ovex 6555 . . . 4 (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V
118, 9, 10fvmpt 6176 . . 3 (𝑅 ∈ V → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
12 fvprc 6082 . . . 4 𝑅 ∈ V → (oppr𝑅) = ∅)
13 reldmsets 15664 . . . . 5 Rel dom sSet
1413ovprc1 6560 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) = ∅)
1512, 14eqtr4d 2646 . . 3 𝑅 ∈ V → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
1611, 15pm2.61i 174 . 2 (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
171, 16eqtri 2631 1 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1474  wcel 1976  Vcvv 3172  c0 3873  cop 4130  cfv 5790  (class class class)co 6527  tpos ctpos 7215  ndxcnx 15638   sSet csts 15639  Basecbs 15641  .rcmulr 15715  opprcoppr 18391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-res 5040  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-tpos 7216  df-sets 15647  df-oppr 18392
This theorem is referenced by:  opprmulfval  18394  opprlem  18397
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